Pattern formation of reaction-diffusion system with chemotaxis terms

In this paper, we systematically study two-species reaction-diffusion system with chemotaxis terms. We, first, compare conditions for chemotaxis-driven instability and Turing instability. It follows that conditions for chemotaxis-driven instability are the generalization of conditions for Turing instability without chemotaxis. Most of all, we provide sufficient conditions for chemotaxis-driven instability, which implies that chemotaxis can give rise to pattern formation for the same diffusion coefficients. To support our theoretical analyses, we take the Rosenzweig-MacArthur model as an example to illustrate the influence of parameters on pattern formation. By conditions for chemotaxis-driven instability and numerical simulations, we show parameter spaces of chemotaxis-driven instability (Turing spaces). In addition, we establish the similarity and difference between these Turing spaces. Our numerical simulations validate the dependence of pattern formation on parameters and that unstable parameter spaces induced by chemotaxis can be sufficiently larger than that induced by the reaction-diffusion system without chemotaxis (standard Turing space). Furthermore, we present the pattern formation induced by chemotaxis for D_u=D_v. For numerical simulations, we can choose r and β from the Turing spaces to validate previous analysis.